FDTD-Based Simulation and Analysis of Noise Reduction Effects in Hospital Wards Yiquan Xu 1 , Hequn Min 2 Key Laboratory of Urban and Architectural Heritage Conservation, Ministry of Education, School of Architecture, Southeast University, 2 Sipailou, Nanjing 210096, China

ABSTRACT The impact of domestic hospital acoustic environment on the health of patients and healthcare workers is attracting more and more attention from all walks of life. In this paper, we investigated and analyzed the noise situation of typical double person wards in the hospital based on the finite difference time domain method (FDTD), and designed two noise reduction schemes according to common noise sources occurring in the hospital twin room. Sound fields in double wards were numerically simulated before and after noise reduction scheme treatments applied, for evaluation on noise reduction effects from two schemes. Results show that, the FDTD method can be used to effectively model sound fields in hospital wards with complex boundary conditions like surfaces and shapes of standard indoor facilities; meanwhile, the proposed noise mitigation treatments have significant effects on improvement of reverberation time and speech intelligibility in wards.

1. INTRODUCTION

Since the end of last century, more and more researchers have gradually started to pay attention to the acoustic environment in medical settings, especially negative effects of adverse acoustic environments on health care workers and patients [1]. Noise can affect both human physiological and psychological health [2], and studies have shown that long-term exposure to a noisy environment can hamper wound healing after surgery [3] and prolong hospitalization [4] and that substantial amounts of noise can also impede voice communication and interfere between the work of health care [5]. As the finite-difference time-domain (FDTD) method can directly solve fluctuation equations for indoor acoustic environments with complex boundary conditions and produce consistent calculation results with accuracy over a wide range of frequency bands, in this paper, we will use the FDTD method to simulate sound fields and conduct noise reduction treatments in hospital wards in actual situations. 2. Theory on FDTD-based sound field modeling

The mathematical basis of the FDTD method is Maxwell ’ s equations [6], which discretize fluctuation equations in the time domain and step through calculations to obtain the time-domain signal waveform at points in space. The FDTD method was first applied to acoustic modeling in the late nineties [7] and can now satisfy sound field simulation calculations under different boundary

conditions [9]. Based on Bilbao ’ s study, Hamilton detailed discussed [10] the application of FDTD

1 yqxu0919@126.com 2 hqmin@seu.edu.cn

modeling in indoor acoustics for different environments. In this paper, modeling and simulation based on the work of Hamilton [10] and focusing on the recognition and calculation of boundaries of models, further discusses the modeling approach in complex interior environments. And in this project, many objects needed to be added to model to simulate the real hospital environment. It is feasible to process each item once in the program in simple model testing because the workload is not significant. However, when the model becomes more complex like the real room inside, it would lead to tedious work, and a higher risk of errors would affect the simulation effect and experimental results. Therefore, we optimized the construction part of the model through an automated process as shown in Figure 1 and further details will be discussed later.

Figure 1: Pre-processing of Obstacle Boundary Identification in the Model.

2.1. FDTD Method

The numerical simulation starts with the acoustic wave propagation, describes the transmission behavior of an ideal sound in the medium using the PDE equation, and approximates the numerical solution of the PDE equation using the FDTD method, where time and space domains within the acoustic model are uniformly discretized and distributed over grid points in a Cartesian coordinate system. In the discrete system, the sound pressure is calculated for each point and the continuous fluctuation equation is transformed into the DPDE equation [9], applying FDTD operators to obtain:

1

2 𝑝−∆𝑝= 0 , (1)

𝑐 2 𝜕 𝑡

Model design Object process Build model ‘Add object information Process and caleulate the index of the object in the model, and check the relationship between the object and the boundary Not _‘botertnn whether objec? attached [Set up ghost nodes of

𝑛+1 = 𝜆 2 𝑄 𝑙,𝑚,𝑞

𝑛−1 , (2)

𝑛 + (2 −6𝜆 2 )𝑝 𝑙,𝑚,𝑞

𝑛 −𝑝 𝑙,𝑚,𝑞

𝑝 𝑙,𝑚,𝑞

Where:

𝑛 = 𝑝 𝑙+1,𝑚,𝑞

𝑛 + 𝑝 𝑙−1,𝑚,𝑞

𝑛 + 𝑝 𝑙,𝑚+1,𝑞

𝑛 + 𝑝 𝑙,𝑚−1,𝑞

𝑛 + 𝑝 𝑙,𝑚,𝑞+1

𝑛 + 𝑝 𝑙,𝑚,𝑞−1

𝑄 𝑙,𝑚,𝑞

𝑛 , (3)

𝜆= 𝑐𝑇

ℎ (4)

q m l         refers to sound

Here c is the sound velocity and ) , , , ( , , T n t h q z h m y h l x p n

pressure elements in discrete system, R h  is the space between grid points of discrete system and

s F T 1  ). q m l , , are space indexes in different

T is the time step of system ( sampling frequency

directions, and n is the index in time, and Z n q m l  , , , . And  is a dimensionless constant called Courant number, which means that at each time step, the sound wave will be further propagate through the  grid unit. Furthermore, it will be limited by stability considerations.

2.2. Boundary Elements Marking Method for Acoustic Models

Boundaries in the model are set to be locally responsive, the normal component of the velocity of point on boundary surface is only related to the preceding sound pressure element, and neighboring elements are not affected. The boundary condition processes in this paper are based on the study of sound pressure elements on the boundary in [10], and the propagation of acoustic pressure occurs only within the model boundary. By using the six-point update method, when the sound pressure element is on boundary, some adjacent elements out of range are defined as ‘ghost node’ and their sound pressure value are set to zero, Figure 2 indicates how sound pressure will be calculated in different position, neighboring units with black dot mean they are still in the model’s range and

those units with marker “ × ” are out of range.

Figure 2: Corner, Edge, Surface and Inner Elements of Model. Based on the sound pressure calculation method proposed by Hamilton [10], a spatial model labeling approach is conceived to label the spatial location of each element inside the model to the boundary, defining classification factors K as the number of ghost nodes for each unit. An alternative representation is used and carried into the model simulation calculation as shown in Figure 3:

𝛫 ′ = 6 −𝛫 , (5)

Figure 3: The corners, edges, surfaces, and inner ghost nodes of the upper left part of the 3D model.

2.3. Numerical Stability and Simulation Accuracy

The value of  generated by FDTD method determine the stability and accuracy of sound field models. The model accuracy is constrained by the FDTD principle, where numerical phase velocity in discrete systems is close to theoretical phase velocity at low frequencies and will be lower than theoretical phase velocity at high frequencies, so limit values used in the simulation are calculated to reduce effects of numerical dispersion while oversampling is used to reduce grid spacing to ensure the accuracy of FDTD method [10].

The simulation experiments in this paper were conducted under MATLAB 2020, using GPU acceleration to transfer computational tasks from the CPU to the GPU for completion and back, reducing the total simulation computation cost from hours to minutes, with a maximum GPU utilization of 94% during the acceleration process. 3. Sound Field Simulations of model with complex indoor boundary conditions

The dual-person ward acoustic environment study compares results before and after noise reduction treatment using reverberation time and speech intelligibility as indicators. The main noise causes in hospital wards are human voices and instrument working sound and the quality of acoustic environment can be improved by adding noise reduction facilities. Referring to the national

medical architectural standards, the typical two-person ward size is set at 4.75m × 3m × 3m and is

routinely arranged with two beds, two sub-cabinets, a door, glass windows, few cabinets, and five lighting lamps, and the model structure is shown in the black frame in Figure 4.

Surfaces of each object inside the model can be considered as boundaries in the model and have different absorption and reflection coefficients, but these objects are located inside the model and occupy a certain volume, so the sound waves cannot directly penetrate these objects without energy losses. As stated above for ghost node, sound pressure out of model’s range are set to be zero, and here the interior points of objects can also be considered as points outside the range.

Figure 4: Structure of FDTD acoustic model.

The location property of markers 𝛫 ′ is used in the calculation process to determine the currently updated sound pressure element as a surface, edge, or corner point in the model; after releasing excitation at the sound source position, the direction of motion at each element is judged during calculation. This allows flexible placement of objects with different absorption coefficients within the model or changes to the interior structure, decomposing all obstacles into elements; the fluctuation equation is combined with the element type determination in FDTD method to obtain sound pressure value at each point. The simulation results of sound field under complex structure, i.e., the propagation and reflection of sound pressure are shown in the cross-section in Figure 5.

Figure 5: Structure of FDTD acoustic model and propagation of sound pressure at different times.

To design noise reduction experiments, two materials were selected for simulation in the model by laying highly absorbent materials on the ceiling or the wall at the bedside to see effects on the acoustic environment after adding absorbent materials on ceiling and wall respectively, and Table 1 shows the absorption coefficients of materials tested. The target frequency range of simulation experiments is 125 Hz-4 kHz, which covers the normal hearing range. In the experiment, a sound source is set near patient beds to simulate the working position of medical staff, and receiving

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points are placed above beds to imitate the patient ’ s bed-lying state.

Table 1: Sound absorption coefficient of wall and ceiling, Frequency range:125Hz-4kHz.

Sound-absorbing materials 125Hz 250Hz 500Hz 1kHz 2kHz 4kHz

Ori Ceiling 0.50 0.50 0.50 0.50 0.50 0.50

Ceiling A 0.55 0.80 0.85 0.90 1.0 0.95

Ceiling B 0.45 0.90 0.95 0.85 0.95 0.90

Ori Wall 0.365 0.365 0.365 0.365 0.365 0.365

wiradation duaetion = C.225000 (shiner 1200

Wall A 0.20 0.75 1.0 1.0 1.0 1.0

Wall B 0.25 0.80 0.95 1.0 1.0 1.0

4. COMPARISON AND ANALYSIS OF SOUND FIELD SIMULATION RESULTS

4.1. Optimized Solutions for Acoustical Absorption of Ceilings

In the initial model, ordinary acoustic panels ‘Ori Ceiling’ were used at the ceiling. And in comparison of schemes, highly absorbent materials ‘Ceiling A’ and ‘Ceiling B’ were used. Results

of simulations and theoretical experiments are shown in Figure 6. The hospital ward’s 𝐸𝐷𝑇 slightly decreased in 250Hz-1kHz band after using the new material, but increased and even turned out to be higher in 2kHz-4kHz band than 𝐸𝐷𝑇 ’s value before the adjustment. Experimental models’ 𝑇 20 were lower than original model in 125-250Hz bands, but got to be higher than the original model from the 500Hz band. And more significant differences shown in 𝑇 30 's comparison, after applying new materials into model, the value of 𝑇 30 raised sharply with the increase of frequency and much higher than the initial value in all frequency bands. These experimental results indicate that noise reduction effect obtained by strong acoustic materials were not satisfactory and had limited effect on the enhancement of speech intelligibility. In general, the change in ceiling was not significant for improvement of hospital ward’s sound environment, but it can play a certain role in some frequency bands.

Figure 6: 𝐸𝐷𝑇 ， 𝑇 20 ， 𝑇 30 ， 𝐶 50 of ceiling optimization scheme.

4.2. Optimized solution for absorption of sound on walls

The wall material in the initial model was latex paint coating, and two medical acoustic panels ‘Wall A’ and ‘Wall B’ were selected for the noise reduction scheme. The theoretical experimental results are shown in Figure 7. 𝐸𝐷𝑇 in both new models were below 0.35s in all frequency bands. The trends of 𝑇 20 with frequency were similar for both new models, and 𝑇 20 ’s value were lower than that of the initial model in the vast majority of frequency bands. In model 1, 𝑇 30 was higher than the initial model after the frequency exceeds 1 kHz. While in model 2, 𝑇 30 maintained stable performance with increasing frequency and its value was lower than the initial model. The intelligibility of indoor speech was improved, compared with initial hospital ward in all frequency bands except 125Hz band. This indicates that in the initial model ‘Ori Wall’ reflects most of the sound waves near the boundary, while the acoustic panel absorbs most of the sound waves traveling to the boundary, reducing impacts of reflected sound on receiving points. The results obtained through simulation experiments show that the wall with acoustic panels has good effects in both reducing the reverberation time and improving the speech intelligibility.

Figure 7: 𝐸𝐷𝑇 ， 𝑇 20 ， 𝑇 30 ， 𝐶 50 of wall optimization scheme.

5. CONCLUSION

In this paper, we investigated and analyzed the noise situation of typical twin ward of hospital, and designed two noise reduction plans according to main noise sources and simulated sound fields before and after implementation of noise reduction measures with the FDTD method. In the process of FDTD calculation, this paper designed a method that can identify boundaries of obstacles flexibly within the model and build basic interior frames for sound field modeling. Sound fields of twin wards were numerically simulated with the original layout and two proposed noise mitigation schemes, and further compared and analyzed noise reduction effects of ceiling and wall side of hospital ward after acoustic treatment respectively. As simulation results show, the FDTD method can be applied to effectively build sound field models for hospital wards with many obstacles and complex boundaries such as standard indoor facilities placed inside. Meanwhile, numerical experiments outcomes show that both reverberation time and speech intelligibility in the ward are more obviously improved after taking the noise reduction treatment on the wall side. Field experiments could not be conducted at this stage of the project due to epidemic restrictions, and further studies on optimization of ward sound environment will be conducted in terms of both medical acoustic noise reduction materials and room layouts in wards and other locations with severe noise conditions. 6. REFERENCES

[1] E. Ryherd, J. E. West, I. J. Busch-Vishniac, and K. P. Waye, Evaluating the hospital

soundscape, Acoustics Today, vol. 4, no. 4 , pp. 22 – 29, 2008. [2] W. Passchier-Vermeer and W. F. Passchier, Noise exposure and public health. Environmental

health perspectives, vol. 108, no. suppl 1 , pp. 123 – 131, 2000.. [3] B. B. Minckley, A study of noise and its relationship to patient discomfort in the recovery room,

Nursing Research, vol. 17, no. 3, pp. 247 – 249, 1968.

[4] D. Fife and E. Rappaport, Noise and hospital stay. American Journal of Public Health, vol. 66,

no. 7, pp. 680 – 681, 1976.. [5] B. Berglund, T. Lindvall, D.H. Schwela, W. H. Organization et al.,Guidelines for community

noise, 1999.

[6] K. Yee, Numerical solution of initial boundary value problems involving maxwell ’ s equations

in isotropic media, IEEE Transactions on antennas and propagation, vol. 14, no. 3, pp. 302 – 307, 1966. [7] L. Savioja, Simulation of room acoustics with a 3-D finite difference mesh . Ann Arbor, MI:

Michigan Publishing, University of Michigan Library, 1994. [8] D. Botteldooren, Finite-difference time-domain simulation of low-frequency room acoustic

problems, The Journal of the Acoustical Society of America, vol. 98, no. 6, pp. 3302 – 3308, 1995. [9] S. D. Bilbao, Numerical sound synthesis. Wiley Online Library, 2009. [10] B. Hamilton, Finite difference and finite volume methods for wave-based modelling of room

acoustics , 2016.